Mathcad Community Challenge May 2026: Starman Star Geometry for the Greatest Rock Band of All Time!

This month’s challenge is inspired by the greatest rock band of all time: Rush! Specifically, it is in honor of the 50th anniversary of their concept album 2112, which gave us the Starman logo. Here you can see the logo along with geometry in Creo pertaining to the challenge:

Note that the Starman logo is not a pentagram, and Rush is not devil-worshipping music (as many of our parents thought). Drummer and lyricist Neil Peart (RIP) was really into Ayn Rand (which is arguably worse). The logo represents a naked man, armed only with his intellect, resisting the collectivism of the Solar Federation. Listen to the album. It’s awesome.

Here is the challenge: start with a circle of radius 1. Inscribe a regular pentagon. Connect the vertices to form a star.

Challenge 1: Measure the area of (1) the star, (2) the pentagon inside of the star, and (3) the tips of the star (the star minus the interior pentagon).

Challenge 2: Write functions for these areas in terms of the original circle radius r.

Challenge 3: Inside of the first star is another regular pentagon. Repeat the process: connect the vertices to form another star. Measure the same three areas as in challenge 1.

Challenge 4: Find the limit of the summed areas as you continue inscribing stars inside of pentagons.

Challenge 5: Graph the limits of these areas on an XY Plot or Chart Component. On the x-axis would be the “level” or “depth” of the inscribed star. On the y-axis would be area of either the summed area, the area at that “level,” or both.

As always, have fun with this! Find the Mathcad Community Challenge guidelines here!